Poisson Distribution Approximated by Hat-Check Distribution
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Theorem
Let $X$ be a discrete random variable which has the hat-check distribution with parameter $n$.
Then $X$ can be approximated by a Poisson distribution with parameter $\lambda = 1$.
Proof
Let $X$ be as described.
Let $k \ge 0$ be fixed.
Then:
\(\ds \lim_{n \mathop \to \infty} \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \dfrac 1 {\paren {n - \paren {n - k} }!} \sum_{s \mathop = 0}^{n - k} \dfrac {\paren {-1}^s} {s!}\) | setting $k = n - k$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {1^k} {k!} e^{-1}\) | Taylor Series Expansion for Exponential Function and Definition of Poisson Distribution |
Hence the result.
$\blacksquare$
Examples
Example: $N = 8$
Let $X$ be a discrete random variable which has the hat-check distribution with parameter $n = 8$.
Then $X$ can be approximated by a Poisson distribution with parameter $\lambda = 1$.